Regularity of n-P-V-Rings and n-P-V’-Rings
The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong <inline-formula><math xmlns="http://www.w3.org/1998/M...
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2024-12-01
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| author | Liuwen Li Wenlin Zou Ying Li |
| author_facet | Liuwen Li Wenlin Zou Ying Li |
| author_sort | Liuwen Li |
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| description | The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is a left n-P-V-ring where all its maximal left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular iff <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an abelian left (right) n-P-V’-ring where all its maximal essential left (right) ideals are either quasi-ideals, weakly right (left) ideals, or generalized weak ideals. (3) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is reduced left weakly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. |
| format | Article |
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| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
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| series | Axioms |
| spelling | doaj-art-0c1f1adf4d5e47d68489aed7cb2b250b2025-08-20T02:55:42ZengMDPI AGAxioms2075-16802024-12-01131286310.3390/axioms13120863Regularity of n-P-V-Rings and n-P-V’-RingsLiuwen Li0Wenlin Zou1Ying Li2School of Science, Jinling Institute of Technology, Nanjing 211169, ChinaSchool of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, ChinaSchool of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, ChinaThe regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is a left n-P-V-ring where all its maximal left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular iff <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an abelian left (right) n-P-V’-ring where all its maximal essential left (right) ideals are either quasi-ideals, weakly right (left) ideals, or generalized weak ideals. (3) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is reduced left weakly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals.https://www.mdpi.com/2075-1680/13/12/863n-P-V-ringsn-P-V’-ringsleft weakly <i>π</i>-regularstrongly <i>π</i>-regularn-P-injective modules |
| spellingShingle | Liuwen Li Wenlin Zou Ying Li Regularity of n-P-V-Rings and n-P-V’-Rings Axioms n-P-V-rings n-P-V’-rings left weakly <i>π</i>-regular strongly <i>π</i>-regular n-P-injective modules |
| title | Regularity of n-P-V-Rings and n-P-V’-Rings |
| title_full | Regularity of n-P-V-Rings and n-P-V’-Rings |
| title_fullStr | Regularity of n-P-V-Rings and n-P-V’-Rings |
| title_full_unstemmed | Regularity of n-P-V-Rings and n-P-V’-Rings |
| title_short | Regularity of n-P-V-Rings and n-P-V’-Rings |
| title_sort | regularity of n p v rings and n p v rings |
| topic | n-P-V-rings n-P-V’-rings left weakly <i>π</i>-regular strongly <i>π</i>-regular n-P-injective modules |
| url | https://www.mdpi.com/2075-1680/13/12/863 |
| work_keys_str_mv | AT liuwenli regularityofnpvringsandnpvrings AT wenlinzou regularityofnpvringsandnpvrings AT yingli regularityofnpvringsandnpvrings |